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Symmetry Properties of Superconducting Order Parameter in Sr2RuO4

A Brief Review

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Abstract

Soon after the discovery of superconductivity in Sr2RuO4 (SRO) a quarter-century ago, it was conjectured that its order parameter (OP) has a form similar to that realized in the superfluid phases of 3-He, namely, odd parity and spin triplet. While the chiral p-wave pairing believed to be realized in the A phase of that system was favored by several early experiments, in particular, the muon spin rotation and the Knight shift measurements published in 1998, the original Knight shift result was called into question in early 2019, raising the question as to whether the “chiral p-wave”, or even the spin-triplet pairing itself, is indeed realized in SRO. In this brief pedagogical review, we will address this question by counterposing the currently accepted results of Knight shift, polarized neutron scattering, spin counterflow half-quantum vortex (HQV), and Josephson experiments, which probe the spin and orbital parts of the OP, respectively, with predictions made both by standard BCS theory and by more general arguments based only on (1) the symmetry of the Hamiltonian including the spin-orbital terms, (2) thermodynamics, and (3) the qualitative experimental features of the material. In the hope of enhancing readers’ intuitive grasp of these arguments, we introduce a notation for triplet states alternative to the more popular “d-vector” one which we believe well suited to SRO. We conclude that the most recent Knight shift and polarized neutron scattering experiments do not exclude in the bulk the odd-parity, spin-triplet “helical” states allowed by the symmetry group of SRO but do exclude the “chiral p-wave”, \({{{\varGamma }}}_{5}^{-}\) state. On the other hand, the Josephson and in-plane-magnetic-field stabilized HQV experiments showed that the pairing symmetry in SRO cannot be of the even-parity, spin-singlet type, and furthermore, that in the surface region or in samples of mesoscopic size the d-vector must be along the c axis, thus excluding all bulk p-wave states except \({{{\varGamma }}}_{5}^{-}\). Possible resolution of this rather glaring prima facie contradiction is discussed, taking into account implications of other important experiments on SRO, including that of the muon spin rotation, which are touched upon briefly only towards the end of this article.

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Notes

  1. Here we absorb any terms of the form a3[c∗(+)2c(−)2 + c.c.] into a2 by writing them as 2a3cos(ϕ)|c(+)|2|c(−)|2, where ϕ = 2argc(+)/c(−), minimizing with respect to ϕ and adding the result to the original a2. Since this complication arises only for the improbable case q < 1 we do not pause on it.

  2. Although the standard derivation of these equations nowhere makes any reference to SO interactions, they are not affected by them, since the conversion from real spin to pseudospin, while it may change the effective magnetic moment of the normal-state excitations and hence affect χn, they cannot change the ratio χ(T)/χn for the singlet state. For the triplet states, SO interactions can at most affect, via the “K-S” terms in the effective Hamiltonian, the numerical value of α. (Note that the situation is very different when the SO interaction fails to conserve parity as in a disordered alloy.)

  3. Once one takes into account the small variation in the DOS on the FS or other asymmetries not considered in the original BCS treatment, the quoted result that χ is unchanged from χn for a state which is ESP with respect to the field axis is not strictly true. The second-order transition then splits in a field. Thermodynamics requires a small increase of χ in the superconducting phase, an effect first predicted for superfluid 3-He A by Takagi [33] and confirmed experimentally by Paulson and Wheatley [34] (see also ref. [35], sect. 13). However, an order-of-magnitude estimate shows that this effect, if it indeed occurs in SRO, would be far too small to be seen in current experiments.

  4. It should be noted that, the Knight shift Δω/ωd in a metal is proportional to \({{\varDelta }} H/H = \frac {8 \pi }{3} <|u_{k}(0)|^{2}>_{E_{F}}{\chi _{e}^{S}} \) where uk(0) is the modulating function of the Bloch function, \(<...>_{E_{F}}\) is the average over the Fermi surface, and \({\chi _{e}^{S}}\) is the total spin susceptibility of the electrons (see ref. [37], p. 122). The electron density, \(<|u_{k}(0)|^{2}>_{E_{F}}\), will vary as the strength of the applied magnetic field used to do the measurement is varied. Indeed, experiments revealed strong dependence of the differential susceptibility dM/dH on the applied field [38]. It is unclear how much of this dependence may be due to the dependence of \(<|u_{k}(0)|^{2}>_{E_{F}}\) on the applied field.

  5. We would like to take this opportunity to note a curious and very misleading error in the English translation of ref. [54], in the fifth sentence on p. 397. A correct translation of the Russian original would be “For the tunnel Hamiltonian to be nondiagonal, the spin-orbital (SO) interaction need not necessarily occur in the insulating layer; it needs only be present in the superconductor”, where it is clear that the “it” refers to “the SO interaction”. However, the translator appeared to have interpreted the “it” as referring to “the insulating layer”, and to have then gratuitously made this misunderstanding explicit in the translation. One of us (AJL) thanks Prof. V. B. Geshkenbein for confirming this.

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Acknowledgments

Y.L. would like to thank X. Cai, R. Cava, S-K. Chung, H. Kawano-Furukawa, V. B. Geshkenbein, K. Hasselbach, W. Huang, J. K. Jain, C. Kallin, H-Y. Kee, J. Kirtley, Y. Maeno, K. Moler, T. M. Rice, J. A. Sauls, M. Sigrist, C-C. Tsuei, D. J. van Harlingen, V. Vakaryuk, S-K. Yip and F-C. Zhang for useful discussions over the years and K. Ishida for sharing the new Knight shift data in April 2019 prior to the publication. A.J.L. thanks Dale van Harlingen and Vidya Madhavan for ongoing discussions about exotic superconductivity. Much of the work done at Penn State was supported by DOE under Grant No. DE-FG02-04ER46159.

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Authors and Affiliations

  1. Department of Physics, University of Illinois, Urbana, Illinois, 61801, USA

    Anthony J. Leggett

  2. Shanghai Center for Complex Physics and School of Physics and Astronomy, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240, China

    Anthony J. Leggett

  3. Department of Physics and Materials Research Institute, Pennsylvania State University, University Park, PA, 16802, USA

    Ying Liu

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It is a pleasure to dedicate this paper to Brian Josephson, and to wish him many more happy years of activity in physics and elsewhere.

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Appendices

Appendix 1.: The d-vector notation

The following is adapted from ref. [35], section VII.C. To begin with, the d-vector notation was originally introduced into the superconductivity literature by Balian and Werthamer [81]; it is very convenient for discussion of the rotational properties of triplet states, and has been very widely used in the subsequent literature.

Consider an arbitrary symmetric (but not necessarily Hermitian) 2 × 2 matrix \(\hat Q\) with matrix elements Qαβ. We can form a vector \(\hat {\mathbf {Q}}\) (whose Cartesian components are in general complex) by the prescription

$$ Q \equiv -\frac{1}{2} i \sum\limits_{\alpha \beta}(\sigma_{2} {\boldsymbol{\sigma}})_{\alpha \beta}Q_{\alpha \beta}, $$
(A1.1)

where the σi (i = 1, 2, 3) are the standard Pauli matrices. Inversion of Eq. A1.1 gives after a little algebra:

$$ Q_{\alpha\beta} = i \sum\limits_{i=1}^{3}(\sigma_{i} \sigma_{2})_{\beta\alpha} Q_{i} \equiv i \sum\limits^{3}_{i=1}(\sigma_{i} \sigma_{2})_{\alpha\beta} Q_{i}, $$
(A1.2)

or more explicitly, matrix \(\hat Q\) has the form,

$$ \left( \begin{array}{ccc} -Q_{x}+iQ_{y} & Q_{z} \\ Q_{z} & Q_{x}+iQ_{y} \end{array} \right) $$
(A1.3)

We note the relation (valid for arbitrary \(\hat Q\))

$$ \frac{1}{2} Tr\hat{Q}\hat{Q}^{\dagger} \equiv \vert Q \vert^{2}. $$
(A1.4)

In the text, we shall choose the quantity Q(r1,r2 : σ1,σ2) to be the normalized pair wave function, i.e. the quantity F(r1,r2 : σ1,σ2) multiplied by \(1/\sqrt {N} \) where N is its modulus squared summed over the sigmas and integrated over the r’s. We relabel the vector Q as d. (In the literature, d(r) (or more usually its Fourier transform d(k)) is frequently identified with the components of the matrix energy gap (or “off-diagonal field”) Δαβ(k). But we do not wish to do this since in a general (not necessarily BCS-like) description this quantity is not necessarily defined). Except in Section 5, we will mostly be interested in the case where F is independent of the COM coordinate; in that case it follows straightforwardly from Eq. A1.4 that the integral of Tr(d(r) ⋅d‡(r)) over r (or of Tr(d(k) ⋅d‡(k)) over k) is unity.

We list some properties of the vector d(r,R):

  1. 1.

    Its squared magnitude |d(r,R)|2 = N− 1|F(r,R)|2 is clearly a measure of the probability density of Cooper pairs with COM position R and relative coordinate r (note that this quantity, even when integrated over r, is not directly related to the total electron density). In the following, we omit a possible dependence of d on R.

  2. 2.

    For a unitary state,which in this notation is defined by the condition that all components of the vector d(r) are real up to an overall multiplying complex constant f(r), this vector can be identified with a direction in spin space, which from inspection of Eq. A1.3 has the property that the operator relation

    $${\mathbf{d}}(\mathbf{r}) \cdot \mathbf{S}_{pair} F(\mathbf{r}, \sigma_{1}, \sigma_{2}) = 0 $$

    is satisfied, where Spair is the vector operator with Cartesian components Si within the triplet spin manifold, i.e., it is the vector such that a pair of electrons formed with relative coordinate r has spin equal to zero along direction d(r).

  3. 3.

    A unitary state has the property that for given r, then with respect to any axis perpendicular to d(r) the pairing is only of parallel spins (+ + and −−), with equal amplitude (but in general different phases). For such a state, there is not even a “local” value of the Cooper pair spin n ⋅Spair along any axis n.

  4. 4.

    If a unitary state has the property that the vector d(r) is independent of r, then it also has the “ESP” property (which is shared by some nonunitary states, see (5) below), namely, it is possible to choose an r-independent basis such that pairing is only of + + and −− with respect to the z axis, in this case with equal amplitude.

  5. 5.

    Finally, a brief note on nonunitary states (which are mostly not of interest in the main text): These have the property that the vector d(r) is nontrivially complex. In this case it is still possible, for any given r, to find an axis n such that n ⋅Spair = 0, and also to find a choice such that only + + and −− pairs are formed, but in this case not with equal amplitude; such a state will in general have a nonzero local pair spin polarization. In the special case that the phase of d(r) (but not necessarily its amplitude) is independent of r, it is a matter of definition, which we do not really need to decide for present purposes, whether we call it “ESP” or not; we choose to do so. The A1 phase of superfluid 3-He is believed to be an extreme case of such a phase with pairing only among + + spins; the A2 phase (the A phase in nonzero magnetic field) is an example of the more general nonunitary case.

The conversion from the d-vector notation to the “S-K” one used in the body of this paper is straightforward provided that the OP is even under reflection in the ab plane (so that the awkward Kz = 0 state does not occur): we simply use Eq. A1.3 and put (kx ±iky) or (x ±iy) \(\rightarrow K(+/-) \). Similarly we enable the reverse conversion by using the reverse transformation in conjunction with Eq. A1.1.

Appendix 2.: Zeeman substates of the spin triplet

To save the reader a little time, in this appendix we list some transformation properties of the Zeeman substates of the spin triplet state under rotation. For clarity we denote the Zeeman substates corresponding to S = 1 and Sz = + 1, 0,− 1 by |S(+) >, |S(0) >, and |S(−) >, respectively, and a spin rotation around the axis n by an angle 𝜃 in Rn(𝜃). In the following we list a few special cases which will be useful for analysis in sects. 2 and 4.

$$R_{z}(\pi)|S(+)> = -|S(+)> $$
$$R_{x}(\pi)|S(+)> = -|S(-)> $$
$$R_{y}(\pi) = R_{x}(\pi) $$
$$R_{z}(\pi)|S(0)> = +|S(0)> $$
$$R_{x}(\pi)|S(0)> = -|S(0)> $$
$$R_{z}(\pi)|S(-1)> = -|S(-1)> $$
$$R_{x}(\pi)|S(-)> = -|S(+)> $$
$$R_{z}(\pi/2)|S(+)> = i|S(+)> $$
$$R_{x}(\pi/2)|S(+)> = \frac{1}{2}(|S(+)> + |S(-)>) + i\frac{1}{\sqrt{2}}|S(0)> $$
$$R_{z}(\pi/2)|S(0)> = |S(0)> $$
$$R_{x}(\pi/2)|S(0)> = -i\sqrt{2}(|S(+)> - |S(-)>) $$
$$R_{z}(\pi/2)|S(-)> = -i|S(-)> $$
$$R_{x}(\pi/2)|S(-)> = \frac{1}{2}(|S(+)>+|S(-)>) - i\frac{1}{\sqrt{2}}|S(0)> $$

Thus, if we consider the \({{{\varGamma }}}_{5}^{-}\) state and rotate the spin coordinates (only) into the ab plane by (say) Rx(π/2), it will come out as a fully ESP state with respect to the y axis (and the same follows by symmetry for any pair of orthogonal axes in the xy plane); if we do the same to any of the helical states \({{{\varGamma }}}_{1-4}^{-}\) we will get a state which with respect to the y axis is half ESP and half n ⋅S = 0; this result was used in Section 4.3. (Needless to say, use of the d-vector notation streamlines the formal calculation, but may be less intuitive.)

Finally, a result we shall use in Section 5.1: Consider the operator f1σ1 + f2σ2 acting on the singlet state of spins 1 and 2. The part that is even in the exchange of 1 and 2 just returns this state, while the odd part is proportional to

$$(f_{1} - f_{2})({\boldsymbol{\sigma}}_{1} - {\boldsymbol{\sigma}}_{2})|singlet> = (f_{1} - f_{2})|{\mathbf{d}}> $$

where |d > is the triplet specified by d; the validity of the equality is easily checked by inspecting the (Cartesian) z component and using rotational invariance.

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Leggett, A.J., Liu, Y. Symmetry Properties of Superconducting Order Parameter in Sr2RuO4. J Supercond Nov Magn 34, 1647–1673 (2021). https://doi.org/10.1007/s10948-020-05717-6

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